BackDetermining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 1 + (0.9)ⁿ
Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.
a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (-4)ⁿ/n!
Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.
Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.
If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]